PORTUGALIAE MATHEMATICA
Vol. 60 Fasc. 3 – 2003
Nova Série
HYPERSURFACES OF INFINITE DIMENSIONAL
BANACH SPACES, BERTINI THEOREMS AND
EMBEDDINGS OF PROJECTIVE SPACES
E. Ballico *
Abstract: Let V , E be infinite dimensional Banach spaces, P(V ) the projective
space of all one-dimensional linear subspaces of V , W a finite codimensional closed linear
subspace of P(V ) and X ⊂ P(V ) a closed analytic subset of finite codimension such that
P(W ) ⊂ X and X is not a linear subspace of P(V ). Here we show that X is singular
at some point of P(W ). We also prove that any closed embedding j : P(V ) → P(E)
with j(P(V )) finite codimensional analytic subset of P(E) is a linear isomorphism onto
a finite codimensional closed linear subspace of P(E).
1 – Introduction
For any locally convex and Hausdorff complex topological vector space V
let P(V ) be the projective space of all one-dimensional linear subspaces of V .
In section 2 we will prove the following result.
Theorem 1. Let V be an infinite dimensional complex Banach space, W a
finite codimensional closed linear subspace of V and X ⊂ P(V ) a finite codimensional closed analitic subset such that M : = P(W ) ⊆ X. Assume that X is not
a linear subspace of P(V ). Then X is singular and its singular locus Sing(X)
contains a closed finite codimensional analytic subset T of M .
By [6], Th. III.3.1.1, Sing(X) is a closed analytic subset of P(V ).
As a very easy corollary of Theorem 1 we will prove the following result.
Received : September 30, 2002; Revised : October 29, 2002.
AMS Subject Classification: 32K05, 58B12, 14N05.
Keywords and Phrases: infinite-dimensional projective space; Banach analytic set; Banach
analytic manifold; singular Banach analytic set; Berini theorem.
* The author was partially supported by MURST and GNSAGA of INdAM (Italy).
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E. BALLICO
Proposition 1. Let V be an infinite dimensional complex Banach space, W a
finite codimensional closed linear subspace of V, X ⊂ P(V ) a finite codimensional
closed analytic subset of P(V ) and Q a degree d hypersurface of P(W ) such that
Q ⊂ X. Then for every integer t such that 0 < t < d every degree t hypersurface
Y of P(V ) containing X is singular at at least one point of Q.
A key point of our proof of Theorem 1 is the following Bertini type result
which will be proved in section 2
Theorem 2. Let V be an infinite dimensional complex Banach space, A a
finite dimensional linear subspace of V and Y a closed analytic hypersurface of
P(V ). Then there exists a linear subspace B of V such that A ⊂ B, dim(B) =
dim(A) + 1 and Sing(Y ) ∩ A = Sing(Y ∩ B) ∩ A.
We believe that Theorem 2 has an independent interest, because it allows quite
often to transfer properties which are known in the case of a finite-dimensional
ambient projective space to the case of finite-codimensional closed submanifolds
of P(V ) with V any Banach space. We used this informal principle to guess the
truth of Theorem 1 and then proved Theorem 2 to prove our guess.
In section we will prove the following classification of all finite codimensional
embeddings of infinite dimensional projective spaces.
Theorem 3. Let V and E be infinite dimensional complex Banach spaces.
Let j : P(V ) → P(E) be a closed embedding with j(P(V )) finite codimensional
closed analytic subset of P(E). Then j is a linear isomorphism onto a finite
codimensional closed linear suspace of P(E).
As far as we know this is the first uniqueness result for finite-codimensional
embeddings. It shows that the assumption of finite-codimensionality is extremely
strong and probably too restrictive.
2 – Proof of Theorems 1 and 2 and of Proposition 1
Proof of Theorem 2: Set m : = dim(A). Let GA be the closed analytic subset of the Grassmannian G(m + 1, V ) of all (m + 1)-dimensional
linear subspaces of V parametrizing the (m + 1)-dimensional linear subspaces
containing A ([2], §2, or [6], p. 89). We have GA ∼
= G(1, V /A) = P(V /A).
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375
Set Z : = P(A)\(P(A) ∩ Sing(Y )). We may assume Z 6= ∅, otherwise the statement of Theorem 2 is vacuosly true. For every P ∈ Z ∩ Y let TP Y ⊂ P(V ) be
the Zariski tangent space of Y at P . Since P ∈ Z, Y is smooth at P . Thus TP Y
is a closed hyperplane of P(V ). Set GA (P ) : = {U ∈ GA : U ⊂ TP Y }. GA (P ) is
a closed analytic subset of GA of codimension m. If B ∈ GA \GA (P ), then B ∩ Y
is smooth at P . Since dim(Z) = m − 1, there is B ∈ GA with B ∈
/ GA (P ) for all
P ∈ Z, i.e. such that Sing(Y ) ∩ A = Sing(Y ∩ B) ∩ A.
Lemma 1. Fix positive integers m, k and d with d ≥ 2 and 2 k ≥ m > k.
Let Y ⊂ Pm be a degree d hypersurface containing a dimension k linear subspace
L of Pm . Then Sing(Y ) ∩ L 6= ∅.
Proof: Take homogeneous coordinates x0 , . . . , xm of Pm such that L =
{xk+1 = . . . = xm = 0}. Let F be a degree d homogeneous equation of Y . Since
L ⊂ Y , there are degree d − 1 homogeneous polynomials Gi , k + 1 ≤ i ≤ m, such
P
that F = m
i=k+1 xi Gi . Since d − 1 > 0, the polynomials Gi , k + 1 ≤ i ≤ m, are
not constant. Since m − k ≥ k, the restriction to L of the m − k homogeneous
polynomial Gi , k + 1 ≤ i ≤ m, must have at least one common zero, P . At P
every partial derivative ∂F/∂xi , 0 ≤ i ≤ m, vanishes. Hence Y is singular at P .
Proof of Theorem 1: Taking a minimal closed linear subspace of P(V )
containing X instead of P(V ) we reduce to the case in which X 6= P(V ) and X
is not contained in any closed hyperplane of P(V ). By [6], Th. III.2.3.1, X is the
zero-locus of finitely many continuous homogeneous polynomials on V , i.e. the
intersection of finitely many closed algebraic hypersurfaces of P(V ). Let Y be
any closed analytic hypersurface of P(V ) containing X. By assumption we have
d : = deg(Y ) > 1.
(a) Here we will check that Sing(Y ) ∩ M 6= ∅ and that Sing(Y ) ∩ M contains
a finite codimensional closed analytic subset T (Y ) of M . Assume that this is
not true. Then for an arbitrary integer n we may find a dimension n projective
subspace E of M such that E ∩ Sing(Y ) = ∅. Let a be the codimension of X in
P(V ). Take any integer n ≥ 2 a + 1 and any E as above. Using Theorem 2 we
obtain the existence of a dimension n + a linear subspace N of P(V ) such that
Sing(Y ∩ N ) ∩ E = ∅, contradicting Lemma 1.
(b) Take finitely many closed analytic hypersurfaces Y1 , . . . , Yx such that
X = Y1 ∩ . . . ∩ Yx . By part (a) and the infinite dimensionality of M we have
Sing(Y1 )∩. . .∩Sing(Yx )∩M 6= ∅ and that Sing(Y1 )∩. . .∩Sing(Yx )∩M contains a
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E. BALLICO
finite codimensional closed analytic subset of M . Since Sing(Y1 )∩. . .∩Sing(Yx ) ⊆
Sing(X), we are done.
Proof of Proposition 1: Since t < d, Y contains P(W ). Hence by Theorem 1
Y is singular at each point of a non-empty closed analytic subset B of P(W ) with
finite codimension in P(W ). Since Q ∩ B 6= ∅, we are done.
3 – Proof of Theorem 3
Proposition 2. Let V be a locally convex and Hausdorff complex topological vector space, Y any Hausdorff complex analytic set and C any finite dimensional connected closed analytic subset of P(V ). Assume that C is not a point.
Then there is no holomorphic map φ : P(V ) → Y such that φ|φ−1 (Y \φ(C)) :
φ−1 (Y \φ(C)) → Y \φ(C) is a surjective biholomorphism, while φ(C) is a point,
i.e. there is no contraction φ : P(V ) → Y of C.
Proof: The result is well-known if V is finite dimensional; it follows from
the result quoted at the end of this proof. Hence we may assume V infinite
dimensional. Assume the existence of such a contraction φ. Since φ(C) is finite,
there is an open neighborhood Ω of φ(C) in Y such that the holomorphic functions
on Ω induce an embedding of Ω as a closed analytic subset of an open subset of a
complex topological vector space. Hence U : = φ−1 (Ω) is an open neighborhood
of C in P(V ) such that the holomorphic functions on U separates the points
of U \C. Since C is finite dimensional, the vector space H 0 (C, OC (1)) is finite
dimensional. Hence the linear span hCi of C in P(V ) is finite dimensional. The
holomorphic functions on U ∩ hCi separate distict points of U ∩ hCi\C. Since
U ∩ hCi is a neighborhood of C in the finite dimensional projective space hCi and
C has positive dimension, this is well-known to be false (see [3] and references
therein for stronger statements).
Proof of Theorem 3: By [4], Th. 7.1, for every holomorphic line bundle L
on P(V ) there is a unique integer t such that L ∼
= OP(V ) (t). Let d be the unique
∗
∼
integer such that j (OP(E) (1)) = OP(V ) (d). The line bundle OP(V ) (t) has no
global section if t < 0, it is trivial and with only the costants as global sections
if t = 0, while if t > 0 its global sections are given by the degree t continuous
homogeneous polynomials on V . Thus d > 0. Every closed curve of j(P(V ))
has degree divisible by d. Since every finite codimensional closed analytic subset
of P(E) contains a line ([1], Th. 1.1, or modify the proof of a similar statement
HYPERSURFACES
377
given in [7], Lemma 1.4), we obtain d = 1. This implies that any line of P(V )
is sent isomorphically onto a line of P(E). This implies that for any two points
P , Q of j(P(V )) such that P 6= Q the line spanned by P and Q is contained in
j(P(V )). Thus j(P(V )) is a linear subspace of P, proving the result.
Remark 1. In the statement of Theorems 1 and 2 and of Proposition 1 we
assumed that V is a Banach space and not a more general topological vector
space only because in their proof we quoted [6], p. 89 and Th. III.2.3.1. In the
statement of Theorem 3 we assumed that V is a Banach space only to quote [4],
Th. 7.1.
REFERENCES
[1] Ballico, E. – Lines in algebraic subsets of infinite-dimensional projective spaces,
preprint.
[2] Douady, A. – Le probleme des modules pour les sous-espace analytiques compacts
d’un espace analytique donné, Ann. Inst. Fourier, Grenoble, 16 (1966), 1–95.
[3] Hironaka, H. and Matsumura, H. – Formal functions and formal embeddings,
J. Math. Soc. Japan, 20 (1968), 52–82.
[4] Lempert, L. – The Dolbeaut complex in infinite dimension I, J. Amer. Math. Soc.,
11 (1998), 485–520.
[5] Mazet, P. – Un théoreme d’image directe propre, in: “Séminaire Pierre Lelong
(Analyse année 1972–1973)”, Lecture Notes in Math. 410, Springer, Berlin, 1974,
pp. 107–116.
[6] Ramis, J.-P. – Sous-ensembles analytiques d’une varieté banachique complexe, Erg.
der Math. 53, Springer-Verlag, Berlin – Heidelberg – New York, 1970.
[7] Tyurin, A.N. – Vector bundles of finite rank over infinite varieties, Math. USSR
Izvestija, 10 (1976), 1187–1204.
E. Ballico,
Dept. of Mathematics, University of Trento,
38050 Povo (TN) – ITALY
E-mail: ballico@science.unitn.it